Over the past few decades, the traditional critical path method and its various generalizations have become the most popular technique for managing complex projects. It plays a crucial role in differentiating between critical and non-critical tasks to enhance project schedules. For the first time in the literature, our proposed model implements two algorithms for the study of the critical path method, each addressing an advanced framework in the form of a single-valued triangular neutrosophic. The proposed algorithm 1 utilizes Python to extended Dijkstra’s algorithm under the neutrosophic framework, while the proposed algorithm 2 employs linear programming for optimality checks, which is solved using LINGO. Our comparison with previous research on the critical path method shows that the proposed algorithms are better at dealing with uncertainty, making project schedules more reliable and flexible. The findings lead to the proposed algorithm framework, combined with Python and LINGO, to enhance decision-making and improve the accuracy and efficiency of critical path identification in complex project environments.

In the domain of optimization, linear programming (LP) is recognized as an exceptionally effective method for ensuring the most favorable outcomes. Within the context of LP, the minimum cost flow (MCF) problem is fundamental, with its primary objective being to reduce the transportation costs for a single item moving through a network, under the constraints related to capacity. This network is made up of supply nodes, directed arcs, and demand nodes and each arc has an associated cost and capacity constraint, these factors are certain. However, in practical scenarios, these factors are susceptible to variation due to causal uncertainty. The neutrosophic set theory has surfaced as a challenging approach to tackle the uncertainty that is often encountered in optimization processes. In this manuscript, our primary objective is to address the minimal cost flow (MCF) problem while accounting for the uncertainty inherent in the neutrosophic set. We specifically focus on the cost aspect as SVTN numbers and introduce a new approach based on a customized ranking function handmade for the MCF problem a pioneering endeavor within the field of neutrosophic sets. Additionally, we present numerical example to validate the effectiveness and robustness of our model.

Classical inventory models (IM) serve as quantitative tools for determining the optimal order quantities, timing of orders, and safety stock levels for specific inventory items or item groups. Zadeh (1965. Fuzzy sets. Information and Control, 8, 338-353) introduced fuzzy theory and Dubois and Parade (1988. Fuzzy logic in expert systems: The role of uncertainty management. Fuzzy Sets and Systems, 28, 3-17) presented the study of fuzzy inventory model, which, however, exhibits limitations in effectively handling uncertainty, inaccuracies, and imprecise data. In 1999, Smarandache presented the idea of neutrosophic set theory to handle uncertainty. Using trapezoidal neutrosophic numbers, this study extends the idea of neutrosophic sets to inventory management, concentrating on resolving the uncertainty associated with holding costs, ordering costs, and shortage costs. First time within the literature of the neutrosophic set, our new method not only addresses existing problems but can also tackle other issues that no other authors have successfully resolved so far. Additionally, we conduct a comparative analysis of our proposed model against existing models in this article. Based on this comparative study, our findings assert the superior performance of our proposed model in relation to some of the existing models. In conclusion, we wrap up our research by presenting graphical, logical, and tabular comparisons with the existing methods.

A neutrosophic set is a mathematical framework that extends fuzzy and intuitionistic sets to han-dle indeterminate or contradictory information using three components: truth, falsity, and indeterminacy membership degrees which deal with handling indeterminate, imprecise, and uncertain data, while ExtendedFuzzy Theory extends the standard fuzzy set theory to manage more intricate membership degrees. The pri-mary objective of this review is to thoroughly investigate and summarize the existing literature on trapezoidalneutrosophic environments, particularly focusing on aspects such as the NFMOLP Problems in SVTpN envi-ronments. Ultimately, this comprehensive review article aims to enhance the understanding of the potential ofthese integrated methodologies for effectively presenting decision-making amidst complex and uncertain conditions.

In this work, the authors have considered a physical world concern related to an inventory model under a Neutrosophic environment. We aim to perceive the ideal Total Cost (TC) for the proposed Inventory Management (IM) model without considering the shortage. To achieve this aim, we use trapezoidal Neutrosophic environments. For that purpose, we consider an example to clarify the results of the given model and present computational results. The purpose of the suggested method is to not only solve contemporary numerical problems but also handle new sorts of problems. Finally, we conclude our research by providing graphical, logical, and tabular evaluations with the prevailing methodologies.

Fuzzy, intuitionistic, and neutrosophic sets are the primary focus of the review investigation under the extension concepts. The significance and use of triangular shape for data representation are investigated. For modelling and expressing ambiguous or complex data, the triangle shape is a useful tool due to its simplicity and computational efficiency. Further study involves the tool of the operations research technique i.e., Network Analysis that helps in implementing and developing using the fuzzy extension principle. Critical Path Method (CPM) & Project Evaluation and Review Technique (PERT) plays a major part in the field of network in various decision-making scenarios using the real-life applications. In this comprehensive study, the insights of understanding extended fuzzy using the CPM/PERT under various applications are reviewed and analyzed for the future advancements in making more accurate and optimum results.

This paper presents a thorough assessment and classification of different uncertain environments used by researchers to analyze inventory management(IM) systems across various sectors, such as ABC analysis, Last In, First Out (LIFO), and batch tracking. Moreover, it introduces the concepts of the neutrosophic principle and fuzzy principle in inventory management. it also investigates the difficulties associated with the traditional inventory model. The primary focus of the study lies in inventory management under the Neutrosophic principle, specifically addressing uncertain demand and imprecise data. By shedding light on the potential of neutrosophic principle, this manuscript contributes valuable insights into overcoming the challenges posed by fuzzy models and enhancing decision-making in the realm of inventory control system.

Dijkstra’s algorithm (DA) is a very popular approach for finding the shortest route (SR) in the shortest route problem (SRP). The SRP becomes a challenging and complex problem in real life scenarios. The Fermatean neutrosophic set is a mathematical model that combines Fermatean sets with neutrosophic sets. It can handle the unclear, ambiguous, inconsistent, confusing, and uncertain information that comes from real-world problems. Decision-makers face difficulty accurately determining the precise membership (MG) and non membership levels due to the lack of appropriate data available. The FNS can handle this problem. In this study, we consider the interval FNS to describe the arc weight of a neutrosophic graph (NG). This SRP is called an interval Fermatean neutrosophic shortest route problem (IFNSRP). A modified DA is presented to solve this IFNSRP in an uncertain environment. The effectiveness of the presented method is illustrated with a numerical instance of a neutrosophic network.

The study observation offers a method for project control known as the Neutrosophic Critical Path Method (NCPM) which combines the traditional Critical Path Method (CPM) with neutrosophic sets. Dealing with actual global situations that frequently pose demanding situations for CPM specifically while initiatives involve interconnected operations. The presence of uncertainties creates hurdles during challenge execution, due to incomplete and uncertain data. By illustrating surroundings for instance this research outlines the transition from CPM to NCPM and highlights its advantages. Numerical examples have been implemented for solving NCPM, that discover solutions in the face of uncertainty. Furthermore, the advanced Python software specifically designed for network evaluation permits willpower of critical route lengths. This synthesis complements selection-making in project control that paves the manner for precise management of complicated situations.

This paper considers an inventory model under a Neutrosophic environment. There are many inventory models available, such as the EOQ (Economic Order Quantity) models with and without shortage, price breaks, and more. In this paper, we have considered a model without considering the shortage. Our main objective is to find the optimal total cost under uncertainty. To achieve this objective, we use Trapezoidal Neutrosophic environments. Our proposed method aims to solve numerical challenges and new problem types. Finally, the comparative and sensitive analysis is also included in this manuscript. For that purpose, we consider some examples of uncertainty.

The Linear Programming (LP) model, renowned for its robust modeling capabilities, is a highly effective tool for solving real-world problems and optimizing tasks with specific objective. A key component of the LP model is the Minimum Cost Flow (MCF), which aims to minimize the transportation cost of a single product through a capacitated network. This article focuses on the MCF problem with neutrosophic arc cost, a scenario that often arises in real-world situations due to the complexity and uncertainty of data. The neutrosophic theory plays a crucial role in handling this vagueness. The primary objective of this paper is to find the optimal solution to the neutrosophic MCF problem using SVTN numbers.

In decision-making, linear programming is one of the most useful models for obtaining the optimal solution. A crucial element of the linear programming (LP) model is the minimum cost flow (MCF). The objective of the MCF is to reduce the transportation cost of a single product across a network with capacity constraints. Recently, neutrosophic set theory has become a strong way to deal with the uncertainty that often comes with trying to optimize things. This manuscript explores how neutrosophic set theory can be applied to the MCF problem which has caught the interest of some researchers. The primary objective of this study is threefold: firstly, to tackle the MCF problem considering the uncertainty of the neutrosophic set, focusing especially on the cost. Secondly, to introduce an innovative lexicographical method tailored for the MCF problem, marking a first in the field of neutrosphic sets. Lastly, to combine this new method with a multi-objective optimization approach, improving the way we solve the MCF problem in various ways at once. This thorough method is meant to lead to more detailed and effective ways of solving optimization problems when there is uncertainty. To show how our method works, we will go through some numerical examples related to the MCF problem with cost defined by neutrosophic numbers.

The research article offers an extensive examination of extended fuzzy principles and their practical applications for addressing networking problems. Extended fuzzy principles have gained a significant impact that serve as an act of expansion from crisp and fuzzy logic, in addressing various uncertain environmental conditions. The Critical Path Method (CPM) and Project Evaluation and Review Technique (PERT) have emerged as valuable tools for tackling complex applications of network problems. The main emphasis lies on the discussion of three important extended fuzzy principles: Intuitionistic, Pythagorean, and Neutrosophic, specifically in the context of CPMPERT. Overall, this research article aims to provide valuable insights into the application of extended fuzzy principles, enabling better decision-making under different environmental conditions. The findings contribute to the ongoing development of fuzzy extensions and their potential to overcome challenges in networking.

This paper provides a comprehensive evaluation and categorization of the various uncertain environment employed by researchers and scientists to model and analyze inventory management systems in diverse sectors, including healthcare, supply chain, and routing issues. Additionally, it examines the challenges associated with the classical inventory model and introduces the concepts of fuzzy theory and the extended fuzzy principle in inventory management. The article presents important definitions related to fuzzy theory, including the fuzzy inventory model and its challenges. It also explores the applications of the extended fuzzy principle in real-life problems. The study focuses on inventory management under the extended fuzzy principle (Intuitionistic, Neutrosophic, Pythagorean, and so on), considering uncertain demand and imprecise data. The research contributes to the field by providing insights into the potential of fuzzy theory in overcoming the challenges of classical models and improving decision-making in inventory management.